/*
 * (Illustrative dump file, R. J. Cano, Jul 7 2014)
 * 
 * ...Homework self assignment about the piu&biu script property interpreted for almost any real number.
 * 
 * Dear reader: For more details about this, please refer to the script "cano000gpi.txt" linked/attached together this file.
 * 
*/
                           GP/PARI CALCULATOR Version 2.7.1 (released)
                    i686 running linux (ix86/GMP-5.1.3 kernel) 32-bit version
                         compiled: Jun 25 2014, gcc version 4.8.2 (GCC) 
                                     threading engine: single
                          (readline v5.2 enabled, extended help enabled)

                              Copyright (C) 2000-2014 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY 
WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.

parisize = 900000000, primelimit = 500000
? \p500000
   realprecision = 500008 significant digits (500000 digits displayed)
? b=sqrt(prime(12340)/prime(312))^log(gamma(6))^3;
/*
 * Note: The following is a precomputed integer quantity. Its value is specific
 *       for the initial "b" arbitrarily defined by this script. 
*/
? multiple=358210183285547383585137679390533936546268247250069182023566887727956053188465182617570439201046483;
time = 2min, 7,154 ms.
? for(k=1,1,b+=biu(b)); /* Countdown: 2 */
time = 3min, 47,403 ms.
? b/Pi-multiple
time = 441 ms.
%3 = -0.268541....032969
? for(k=1,1,b+=biu(b)); /* Countdown: 1 */
time = 3min, 47,462 ms.
? b/Pi-multiple
time = 453 ms.
%5 = 0.0171493...81389
? for(k=1,1,b+=biu(b)); /* Countdown: 0 */
time = 3min, 46,502 ms.
? ##
  ***   last result computed in 3min, 46,502 ms.
/*
 * End for the warming up with these three preceding iterations. 
 */
? b/Pi-multiple
time = 447 ms.
%7 = -4.149415...76742418196 E-6
? for(k=1,1,b+=biu(b)); /* #1: Initially matching assumed to 5 digits, first actual growing-up iteration here!... */
time = 3min, 43,885 ms.
? b/Pi-multiple
time = 451 ms.
%9 = 5.8759649...869290295608826 E-17 /* Expected 15 decimal places, got 16 */
? for(k=1,1,b+=biu(b)); /* #2 */
time = 3min, 36,180 ms.
? b/Pi-multiple
time = 450 ms.
%11 = -1.668614790719...293506505040869946940145 E-49 /* Expected 48, got 48 */
? for(k=1,1,b+=biu(b)); /* #3 */
time = 3min, 33,609 ms.
? b/Pi-multiple
time = 454 ms.
%14 = 3.821085592252156...03957273928339732417 E-147 /* Expected 144, got 146 */
? for(k=1,1,b+=biu(b)); /* #4 */
time = 2min, 31,684 ms.
? b/Pi-multiple
time = 455 ms.
%16 = -4.58858517131446......1713167717887694228366 E-440 /* Expected 432, got 440 */
? for(k=1,1,b+=biu(b)); /* #5 */
time = 1min, 14,027 ms.
? b/Pi-multiple
time = 451 ms.
%18 = 7.946115826206380...3866959371303932 E-1319 /* Expected 1317, got 1318 */
? for(k=1,1,b+=biu(b)); /* #6 */
time = 17,043 ms.
? b/Pi-multiple
time = 447 ms.
%20 = -4.12651257649986......65442491967350 E-3955 /* Expected 3954, got 3954 */
? for(k=1,1,b+=biu(b)); /* #7 */
time = 6,632 ms.
? b/Pi-multiple
time = 451 ms.
%22 = 5.779203915263121...807337355194 E-11864 /* Expected 11862, got 11863 */
? for(k=1,1,b+=biu(b)); /* #8 */
time = 3,227 ms.
? b/Pi-multiple
time = 455 ms.
%24 = -1.58753224398205...66022733569446501850 E-35590 /* Expected 35589, got 35589 */
? for(k=1,1,b+=biu(b)); /* #9 */
time = 2,056 ms.
? b/Pi-multiple
time = 450 ms.
%26 = 3.290683882362809...082533544878618438 E-106770 /* Expected 106767, got 106769 */
? for(k=1,1,b+=biu(b)); /* #10 */
time = 1,547 ms.
? b/Pi-multiple
time = 459 ms.
%28 = -2.93073797343849...1212286 E-320309 /* Expected 320307, got 320308 */
? print("End ....12286E-320309");
End ....12286E-320309
/* ---------------------------------------------------------------------------------------------------------------
 * A last extra iteration and verification of the expected result hypothetically after infinitely many
 * iterations executed within an hypothetical endless precision environment.
 * ---------------------------------------------------------------------------------------------------------------
*/
? for(k=1,1,b+=biu(b));
time = 351 ms.
? b/Pi-multiple
time = 450 ms.
%31 = 0.E-499909 /* Interpreted as: "b became Pi"; Actually expecting: 906924, however there would be necessary at least \p 906926 */
? \q
Goodbye.

/*
 * With "\p 906926" 14 iterations suffices to a good "multiple"*Pi approximation got by iterations starting from
 *     the arbitrarily chosen real number: sqrt(prime(12340)/prime(312))^log(gamma(6))^3;
 *
 * Conjecture: It might be another property the need of warming up during three iterations before getting the described
 *             "triplication behavior" (this is, before getting multiplied by 3 the quantity of known correct decimal places).
 * 
 * Note: Once more, the timer decrements behavior is observed.
*/

-----------------------------------------------------------------------------------------------------------------------
-------------------------- Appendix: Repeating the experiment shown above, now with \p 906926 --------------------------
-----------------------------------------------------------------------------------------------------------------------

Note: This time for simplicity, all the 14 iterations were made in a row.

? \p 906926
   realprecision = 906933 significant digits (906926 digits displayed)
? b=sqrt(prime(12340)/prime(312))^log(gamma(6))^3;
time = 4min, 12,670 ms.
? for(k=1,14,b+=biu(b));
time = 1h, 14min, 43,706 ms.
? b/=Pi;
time = 745 ms.
? b-=358210183285547383585137679390533936546268247250069182023566887727956053188465182617570439201046483;
time = 2 ms.
? print(b)
0.E-906834

Explanation: 906834+99=906933, So there are no more available digits as decimal places to represent any difference.

:-) 

(End/EOF)